Question

Answer the following questions about the surface area \(S\) on a surface given by a positive function \(z=f(x, y)\) over a region \(R\) in the \(x y\) -plane. Explain each answer. (a) Is it possible for \(S\) to equal the area of \(R\) ? (b) Can \(S\) be greater than the area of \(R\) ? (c) Can \(S\) be less than the area of \(R\) ?

Short answer

(a) Yes, if the function is a flat plane at \(z = 0\). (b) Yes, for any curves, inclinations, or heights in \(f(x, y)\). (c) No, the surface area will always be equal to or greater than the area of \(R\).

Step-by-step solution

Analyze Possibility of Surface Area Equal to Area of Region R

We need to understand that the surface area \(S\) being equal to the area of \(R\) is only possible if and only if the surface is a completely flat plane at \(z = 0\). That's because if \(z = f(x,y)\) for any point \((x, y)\) in region, the surface area \(S\) will equal to area of region \(R\). So, the answer to part (a) is YES, but for a very specific condition.

Analyze Possibility of Surface Area Greater than Area of Region R

Next, the surface area \(S\) can be greater than the area of \(R\) if the function \(z = f(x,y)\) involves any curves, inclinations, or heights in the region \(R\), since they will contribute more to the surface area. So, for any point \((x, y)\) in \(R\) where \(z \neq 0\), the surface area will be more than the area of region \(R\). This makes the answer to part (b) a YES.

Analyze Possibility of Surface Area Less than Area of Region R

Lastly, the surface area \(S\) can never be less than the area of \(R\). The minimum possible surface area can be equal to \(R\) when the surface is a flat plane, and it increases for any heights or curves. So the answer to part (c) is NO.

Key concepts

Surface Area of a Function

When we talk about the surface area of a function, we're referring to the amount of 'space' that the surface covers. This is analogous to how we might wrap a gift—the wrapping paper needs to cover the entire surface of the box. In mathematical terms, if we have a function of two variables, say, \(z = f(x, y)\), that creates a 3D shape over a particular region \(R\) in the \(xy\)-plane, the surface area is defined as the sum of the areas of all infinitesimally small patches on the surface.

Imagine standing in an art gallery, where the floor represents the region \(R\) and the hanging sculptures represent the surface defined by our function \(f(x, y)\). The surface area would be like the amount of canvas you would need to drape over the sculptures to cover them completely, without any gaps or overlaps. This can be calculated using integral calculus, specifically by taking a double integral over the region \(R\). The integral takes into account all the tiny elements of surface area, no matter how the surface curves or bends above \(R\).

It's important to note that the surface area is always either equal to or larger than the area of region \(R\) itself because a three-dimensional surface can extend into space above region \(R\), adding additional area that isn't projected straight down onto the \(xy\)-plane. However, as pointed out in Step 3 of the exercise solution, it cannot be smaller—just as you can't cover a 3D object with less wrapping paper than the size of its base.

Double Integrals in Calculus

To get a little more technical, double integrals are a tool in calculus that allow us to add up an infinite number of tiny pieces to find the whole. In the context of finding the surface area of a function, double integrals let us combine all those minuscule patches of area we were picturing in the art gallery.

Let's say you're given a region \(R\) in the \(xy\)-plane, the way we calculate the surface area of the associated 3D surface is by setting up a double integral over the region \(R\). The integrand includes a term that accounts for the slope of the surface, because steeper slopes will stretch the surface out, creating more area. This term often involves a square root and the partial derivatives of \( f(x, y) \), which tell us how the surface is sloping at each point.

Working through this double integral—often with the assistance of substitution methods or numerical techniques—gives us the total surface area of the shape. Double integrals do not only apply to surface areas, however. They're used in physics to calculate quantities like mass, charge, and probability, and in economics to account for varying rates of production or consumption over different regions. In each case, the double integral sums up a quantity over an area, just in different contexts and with different functions to integrate.

3D Surfaces in Calculus

Lastly, it's important to understand the concept of 3D surfaces in calculus. These are the shapes created when we graph a function like \(z = f(x, y)\), extending over a region \(R\) in the \(xy\)-plane. These surfaces are not always easy to visualize, as they can twist and curve in complex ways. But the principles of calculus allow us to work with these shapes mathematically, even if we can't easily picture them.

For instance, the surface created by the function might rise and dip like hills and valleys, ripple with waves, or spike sharply like a mountain. The diversity of shapes is nearly infinite, and how these shapes relate to their base region \(R\) is crucial in determining their surface area and other properties.

Understanding the nuances of these 3D surfaces—not just how to calculate their areas or volumes—can be critical in fields ranging from engineering, where surfaces might represent stresses on a material, to meteorology, where they might represent varying atmospheric pressures. By studying the way these surfaces behave over a region, we gain insights into the underlying phenomena the functions represent, making 3D surfaces in calculus a foundational concept in both pure and applied sciences.